Towards The Implicit Bias on Multiclass Separable Data Under Norm Constraints

arXiv cs.LG / 3/25/2026

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Key Points

The paper studies how implicit bias from gradient-based training is shaped by optimization geometry when learning multiclass separable data under norm constraints.
It introduces NucGD, a geometry-aware optimizer that uses nuclear-norm constraints to encourage low-rank solution structures.
The work connects NucGD to low-rank projection methods, framing both within a unified perspective on implicit bias and optimization behavior.
To make training scalable, the authors derive an SVD-free parameter update based on asynchronous power iteration.
Experiments analyze how stochastic optimization factors—like mini-batch-induced gradient noise and momentum—affect convergence toward expected maximum-margin solutions.

Abstract

Implicit bias induced by gradient-based algorithms is essential to the generalization of overparameterized models, yet its mechanisms can be subtle. This work leverages the Normalized Steepest Descent} (NSD) framework to investigate how optimization geometry shapes solutions on multiclass separable data. We introduce NucGD, a geometry-aware optimizer designed to enforce low rank structures through nuclear norm constraints. Beyond the algorithm itself, we connect NucGD with emerging low-rank projection methods, providing a unified perspective. To enable scalable training, we derive an efficient SVD-free update rule via asynchronous power iteration. Furthermore, we empirically dissect the impact of stochastic optimization dynamics, characterizing how varying levels of gradient noise induced by mini-batch sampling and momentum modulate the convergence toward the expected maximum margin solutions.Our code is accessible at: https://github.com/Tsokarsic/observing-the-implicit-bias-on-multiclass-seperable-data.